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# Phase Changes

Module by: Paul Padley. E-mail the author

Summary: We look at what happens to the phase of a wave upon reflection.

r ( E 0 r E 0 i ) = n i cos θ i n t cos θ t n i cos θ i + n t cos θ t r ( E 0 r E 0 i ) = n i cos θ i n t cos θ t n i cos θ i + n t cos θ t t ( E 0 t E 0 i ) = 2 n i cos θ i n i cos θ i + n t cos θ t t ( E 0 t E 0 i ) = 2 n i cos θ i n i cos θ i + n t cos θ t

r ( E 0 r E 0 i ) = n t cos θ i n i cos θ t n t cos θ i + n i cos θ t r ( E 0 r E 0 i ) = n t cos θ i n i cos θ t n t cos θ i + n i cos θ t t ( E 0 t E 0 i ) = 2 n i cos θ i n t cos θ i + n i cos θ t t ( E 0 t E 0 i ) = 2 n i cos θ i n t cos θ i + n i cos θ t We can rewrite these equations using Snell's Law to eliminate the cos θ t cos θ t term. From simple trigonometry we know that cos θ t = 1 sin 2 θ t . cos θ t = 1 sin 2 θ t . We also know from Snell's law that sin θ t = n i n t sin θ i sin θ t = n i n t sin θ i so we have cos θ t = 1 n i 2 n t 2 sin 2 θ i . cos θ t = 1 n i 2 n t 2 sin 2 θ i .

We can substitute this into r = n i cos θ i n t 1 n i 2 n t 2 sin 2 θ i n i cos θ i + n t 1 n i 2 n t 2 sin 2 θ i = n i cos θ i n t 2 n i 2 sin 2 θ i n i cos θ i + n t 2 n i 2 sin 2 θ i = cos θ i n t 2 n i 2 sin 2 θ i cos θ i + n t 2 n i 2 sin 2 θ i r = n i cos θ i n t 1 n i 2 n t 2 sin 2 θ i n i cos θ i + n t 1 n i 2 n t 2 sin 2 θ i = n i cos θ i n t 2 n i 2 sin 2 θ i n i cos θ i + n t 2 n i 2 sin 2 θ i = cos θ i n t 2 n i 2 sin 2 θ i cos θ i + n t 2 n i 2 sin 2 θ i

Similarly we can derive that

r = n t 2 n i 2 cos θ i n t 2 n i 2 sin 2 θ i n t 2 n i 2 cos θ i + n t 2 n i 2 sin 2 θ i r = n t 2 n i 2 cos θ i n t 2 n i 2 sin 2 θ i n t 2 n i 2 cos θ i + n t 2 n i 2 sin 2 θ i

t = 2 cos θ i cos θ i + n t 2 n i 2 sin 2 θ i t = 2 cos θ i cos θ i + n t 2 n i 2 sin 2 θ i

t = 2 n t n i cos θ i n t 2 n i 2 cos θ i + n t 2 n i 2 sin 2 θ i t = 2 n t n i cos θ i n t 2 n i 2 cos θ i + n t 2 n i 2 sin 2 θ i

This form allows us to easily plot the coefficients for different values of θ i . θ i . For example,here are the coefficients for the case where n t n i = 1.5 . n t n i = 1.5 .

It is interesting to note that the sign of the coefficient can change on reflection. E r = | r | E = e i π | r | E 0 e i ( K r ω t ) = | r | E 0 e i ( K r ω t + π ) E r = | r | E = e i π | r | E 0 e i ( K r ω t ) = | r | E 0 e i ( K r ω t + π )

This corresponds to a phase change by π π upon reflection.

We can also look at what happens to the reflection coefficients when n i n t = 1.5 . n i n t = 1.5 .

That is going from a high index of refraction material to a lesser. In this case we see at the critical angle we get total internal reflection. What happens to the phase here is complicated.

When we have n i n t > 1 n i n t > 1 (or n t n i < 1 ) n t n i < 1 ) it is convenient to write r = cos θ i i sin 2 θ i n t 2 n i 2 cos θ i + i sin 2 θ i n t 2 n i 2 r = cos θ i i sin 2 θ i n t 2 n i 2 cos θ i + i sin 2 θ i n t 2 n i 2

Now to understand what this implies we need to digress a little. Recall that e i α = cos α + i sin α . e i α = cos α + i sin α . We could have written this as e i α = a + i b e i α = a + i b then we see that α = tan 1 b a . α = tan 1 b a . Now consider cos α i sin α cos α + i sin α = e i α e i α = e 2 i α cos α i sin α cos α + i sin α = e i α e i α = e 2 i α

Now looking back at r r we see that r = e i φ r = e i φ where tan φ 2 = sin 2 θ i n t 2 n i 2 cos θ i . tan φ 2 = sin 2 θ i n t 2 n i 2 cos θ i .

We could go through a similar excersize for r r and get the same result with

tan φ 2 = sin 2 θ i n t 2 n i 2 n t 2 n i 2 cos θ i . tan φ 2 = sin 2 θ i n t 2 n i 2 n t 2 n i 2 cos θ i . Figure 20-8 in Pedrotti and Pedrotti summarizes all the possible phase changes.

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#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

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